Population Proportion

What is the Population Proportion?

A population proportion is a fraction of the population that has a certain characteristic. For example, let’s say you had 1,000 people in the population and 237 of those people have blue eyes. The fraction of people who have blue eyes is 237 out of 1,000, or 237/1000. The letter p is used for the population proportion, so you would write this fact like this:
p = 237/1000.
You can also write 237/100 as a decimal (by dividing 1000 by 237). If you did that, then p = 0.237.

Venn diagram of animals in the clinic, and “dogs”, a subset of the population.

Example question: A veterinary clinic reports that out of 3,412 animals registered at the clinic, 1,712 are dogs, 1,012 are cats and the rest are rodents or birds. What is the population proportion, p, for dogs at the clinic?

Answer: The number of dogs is 1,712 and the total number of animals is 3,412. Therefore, p = 1,712/3,412. As a decimal, that’s p = 1712/3412 = 0.502 (to two decimal places).

Formula

To get “p”, just divide the total population (for the above question, that’s animals in the clinic) by the number of items you’re interested in (in the above case, that’s dogs). As a formula, it’s written as:

p = x / n

Where:
“x” is the number of items you’re interested in, and
“n” is the total number of items in the population.

Note: While “p” is usually used as the symbol for the population proportion, you might also see the letter pi(π) used instead.

Estimating p

In the real world, you usually don’t know facts about the entire population and so you use sample data to estimate p. This sample proportion is written as p̂, pronounced p-hat. It’s calculated in the same way, except you use data from a sample: just divide the total number of items in the sample by the number of items you’re interested in.

Example question: In a survey of 3121 people, 412 are under-vaccinated. What is the proportion of under-vaccinated people in the local population?